The Grunwald Problem for solvable groups

Algebraic Geometry and Number Theory Seminar

Let $K$ be a number field. The Grunwald problem for a finite group (scheme) G/K asks what is the closure of the image of $H^1(K,G) \to \prod_{v \in M_K} H^1(K_v,G)$. For a general $G$, there is a BrauerManin obstruction to the problem, and this is conjectured to be the only one. In 2017, Harpaz and Wittenberg introduced a technique that managed to give a positive answer (BMO is the only one) for supersolvable groups. I will present a new fibration theorem over quasi-trivial tori that, combined with the approach of Harpaz and Wittenberg, gives a positive answer for all solvable groups. Partial results were also obtained independently by Harpaz and Wittenberg.